Method for testing a radio frequency (RF) receiver and related methods

ABSTRACT

A method for testing a radio frequency (RF) receiver may include measuring a plurality of bit error levels for the RF receiver at a given RF frequency. The method may further include applying a Huber function to the measured plurality of bit error levels to generate a bit error ratio (BER) estimate for the RF receiver. The method would also include using the BER estimate to generate a sensitivity for the RF receiver.

FIELD OF THE INVENTION

The present invention relates to the field of communications systems,and, more particularly, to wireless communications systems and relatedmethods.

BACKGROUND OF THE INVENTION

Radio sensitivity measurement plays an important role in evaluating aradio frequency (RF) radio receiver's ability to detect a weak signal ineither a controlled or real application environment. Radio sensitivityand receive antenna gain together determine the total isotropicsensitivity (TIS), which determines the radio downlink performance.

Radio sensitivity is defined as a receiving power level at the input ofthe radio when the bit error ratio (BER) of the radio reaches itsthreshold level. For a Global System for Mobile Communications (GSM)system, a BER of 2.44 is the defined threshold BER level. BERmeasurement accuracy and measurement time can directly affect radiosensitivity measurement accuracy and time.

The relationship of BER and sensitivity is shown in the graph of FIG. 9.Since BER fluctuates significantly in real sensitivity measurements, anaverage value of BER is typically used for estimating the sensitivity ofthe receiver. Yet, due to large spurious noise in the real communicationenvironment and/or the radio itself, and sudden changes in the testenvironment, the average BER may even change significantly.

One exemplary approach for estimating a channel bit error ratio in areceiver is set forth in U.S. Pat. No. 6,792,053 to Vainio et al. Apseudo bit error ratio of a channel is determined in a receivercomprising detecting means for detecting a data sequence of a receivedsignal, decoding means for decoding a first encoding of the detecteddata signal, and re-encoding means for re-encoding with the firstencoding the data sequence decoded from the first encoding. The receiverfurther comprises quality determining means for providing the detecteddata sequence with a value for quality, and estimating means forestimating the bit error ratio-provided that the quality of the detecteddata sequence fulfils a predetermined quality requirement by comparingthe detected data sequence with the data sequence re-encoded with firstencoding.

Despite the existence of such systems, further improvements indetermining or estimating BER in communications systems, particularlywireless communications systems, may be desirable.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram of an exemplary test system fortesting an RF receiver in accordance with one aspect.

FIG. 2 is a schematic block diagram of an alternative test system fortesting an RF receiver.

FIG. 3 is a flow diagram of a method for testing an RF receiver inaccordance with one exemplary aspect.

FIGS. 4-8 are graphs of bit error levels vs. sample numbers and furtherillustrating average BER levels and BER levels obtained using the systemand methods of FIGS. 1-3.

FIG. 9 is a graph of BER vs. normalized TCH level function.

FIG. 10 is a schematic block diagram illustrating exemplary componentsof a mobile wireless communications device for use with the presentinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present description is made with reference to the accompanyingdrawings, in which preferred embodiments are shown. However, manydifferent embodiments may be used, and thus the description should notbe construed as limited to the embodiments set forth herein. Rather,these embodiments are provided so that this disclosure will be thoroughand complete. Like numbers refer to like elements throughout.

Generally speaking, a method is disclosed herein for testing a radiofrequency (RF) receiver. More particularly, the method may includemeasuring a plurality of bit error levels for the RF receiver at a givenRF frequency, and applying a Huber function to the measured plurality ofbit error levels to generate a bit error ratio (BER) estimate for the RFreceiver.

The method may also include using the BER estimate to generate asensitivity for the RF receiver. More particularly, the Huber functionmay be defined as:

${\rho_{k}(f)} = \left\{ \begin{matrix}{f^{2}/2} & {{{if}\mspace{14mu}{f}} \leq k} \\{{k{f}} - {k^{2}/2}} & {{{if}\mspace{14mu}{f}} > k}\end{matrix} \right.$

where k is a positive constant.

Further, k may be defined as:

${k = {2\sqrt{\frac{\sum\limits_{1}^{n}\left( {x_{i} - x_{0}} \right)}{n - 1}}}},$where x₀ is an initial bit error level and n is a total number of biterror levels. Furthermore, the initial bit error level x₀ may be definedas:

$x_{0} = {\frac{\sum\limits_{i}^{n}x_{i}}{n}.}$By way of example, k may be within a range of about 0.8 to 1.4.

Measuring may include measuring the plurality of bit error levels withinan anechoic RF chamber. In other embodiments, the measurements may beperformed in an outdoor environment. The BER estimate may comprise aresidual BER (RBER) estimate, for example. Also by way of example, theRF receiver may be a Global System for Mobile Communications (GSM),General Packet Radio Service (CPRS), and/or an Enhanced Data Rates forGlobal System for Mobile Communications (GSM) Evolution (EDGE) receiver.

In addition, a test system for testing an RF receiver may include an RFsource and a test controller coupled to the RF receiver. Moreparticularly, the test controller may be for measuring a plurality ofbit error levels for the RF receiver based upon transmissions from theRF source at a given RF frequency, and applying a Huber function to themeasured plurality of bit error levels to generate a BER estimate forthe RF receiver.

Referring initially to FIG. 1, a test system 30 for testing an RFreceiver 32, such as a cellular communications receiver, is firstdescribed. The system 30 illustratively includes an RF test source 31coupled to the receiver 32 to be tested via an RF cable 33. By way ofexample, the device receiver 32 may be a Global System for MobileCommunications (GSM) receiver, a General Packet Radio Service (GPRS)receiver, and/or an Enhanced Data Rates for Global System for MobileCommunications (GSM) Evolution (EDGE) receiver, for example. Of course,other suitable wireless receivers may also be used.

In addition, the RF source 31 may be one of a Rohde and Schwartzuniversal radio communication tester CMU 200 or an Agilent 8960 basestation emulator, for example, although other suitable emulators and/orRF test sources may also be used. A test controller 34 is connected tothe device receiver 32 for performing various test operations andmeasurements, which will be discussed in further detail below. It shouldbe noted that while the RF source 31 and test controller 34 areillustrated as separate components in the FIG. 1, the functions of theRF source and test controller may in fact be performed by the same basestation emulator, for example. Alternately, the test controller 34 couldbe a computer or computing device separate from the RF source 31, aswill be appreciated by those skilled in the art.

Turning now to FIG. 2, an alternative test system 30′ is now described.The test system 30′ includes the RF source 31′ (e.g., a base stationemulator), an RF controlled enclosed environment, and the wirelesshandheld device receiver 32′. As will be appreciated by those skilled inthe art, an RF controlled enclosed environment is an electromagnetic(EM) wave shield environment, such as the illustrated EM anechoicchamber 37′ (which may be a full or semi-anechoic chamber), a shieldroom or an RF enclosure. An antenna 35′ connected to the RF source 31′is positioned within the anechoic chamber 37′ and connected to the RFsource 31′ by a coaxial cable to simulate a base station. An antenna 36′for the device receiver 32′ is also positioned within the anechoicchamber 37′ and connected to the receiver.

It should be noted that in typical tests the handheld receiver 32′ andantenna 36′ will be carried by a device housing, but these componentsmay be tested without the device housing if desired. Moreover, theopen-air testing need not be performed in the anechoic chamber 37′ inall embodiments. That is, these test measurements may be made in anoutdoor or actual operating environment.

Various method steps that may be performed by the test controller 32will now generally be described with reference to FIG. 3. As will beappreciated by those skilled in the art, wireless communications devicessuch as cellular devices may operate over one or more frequency bands,each of which in turn includes numerous operating frequencies orchannels. Beginning at Block 40, a plurality of bit error levels aremeasured for the RF receiver 32, at a given one of the RF frequencies,at Block 42. Measurement of bit error levels is well within the skill ofone of ordinary skill in the art using the above-described base stationemulators or other tools, and therefore requires no further discussionhere.

Once the bit error levels for the given RF frequency are measured, thena Huber function is applied to the measured bit error levels to generatea bit error ratio (BER) estimate, such as residual BER (RBER) estimate,for the RF receiver, at Block 44. The BER may then optionally be used indetermining an RF receiver sensitivity of the receiver 32, at Block 46,thus concluding the illustrated method (Block 48). Further details ondetermining receiver sensitivity based upon BER are provided inco-pending application Ser. No. 11/364,999, which is assigned to thepresent Assignee and is hereby incorporated herein in its entirety bereference.

As discussed above, BER fluctuates significantly in actual sensitivitymeasurements, which is why an average value of BER is typically used forestimating the sensitivity of the receiver. Yet, due to large spuriousnoise in the real communication environment and/or the radio itself, andsudden changes in the test environment, the average BER may even changesignificantly. The traditional average used in prior art approaches is aleast square (l₂) method, which is vulnerable to gross errors. That is,if a few spurious data points are present, this can alter the leastsquare average significantly. In order to make this approach more robustagainst gross error, an l₁ method is also sometimes used. However, whenthe data contains many small errors, the l₁ approach can be undesirablybiased towards a subset of the data points.

A Huber function may advantageously be used in accordance with oneaspect to establish a relatively smoother, less biased estimation forBER, which in turn may be used to determine radio sensitivity, as willbe discussed further below. Given measured BER points X=[x₁, x₂, . . . ,x_(n)], the BER that is the best estimation of the measured data pointsis x*, which provides an error function of f_(i)=x_(i)−x*. The Huberfunction is defined as:

$\begin{matrix}{{\rho_{k}(f)} = \left\{ \begin{matrix}{f^{2}/2} & {{{if}\mspace{14mu}{f}} \leq k} \\{{k{f}} - {k^{2}/2}} & {{{if}\mspace{14mu}{f}} > k}\end{matrix} \right.} & (1)\end{matrix}$where k is a positive constant.

The BER may be obtained by solving the following optimization:

$\begin{matrix}{{F\left( x^{*} \right)} = {\overset{n}{\sum\limits_{i}}{{\rho_{k}\left( f_{i} \right)}.}}} & (2)\end{matrix}$The solution for this minimization optimization is different fromtraditional optimization problems which are usually optimizing X. Here,the optimization is finding the value of x* that most accuratelyrepresents the BER. The optimization may be performed iteratively untila minimizer is found having an absolute value less than a giventhreshold or delta, as will be appreciated by those skilled in the art.

In an unbiased data set where no spurious noise is present, x* is equalto the average of all the data points. This point can be used for theinitial point x₀ for the optimization, that is:

$\begin{matrix}{x_{o} = {\frac{\overset{n}{\sum\limits_{i}}x_{i}}{n}.}} & (3)\end{matrix}$The selection of k is an important factor in finding the optimum valueof x*, and may advantageously help speed up the optimization process. Inthe present example k is chosen to be

$\begin{matrix}{k = {2{\sqrt{\frac{\sum\limits_{1}^{n}\left( {x_{i} - x^{0}} \right)}{n - 1}}.}}} & (4)\end{matrix}$Generally speaking, k may be in a range of about 0.8 to 1.4, althoughother values may be used in different embodiments.

With k determined, the data set X can be divided into three subsets,namely:Q={x _(i) |x _(i) −x* ^((m)) |≦k,i=1,2, . . . ,p}P={x _(i) |x _(i) <x* ^((m)) ,i=1,2, . . . ,q}L={x _(i) |x _(i) >x* ^((m)) ,i=1,2, . . . ,l}′  (5)where x*^((m)) is the x* value of m iteration. Furthermore,

$\begin{matrix}{x^{*{({m + 1})}} = {\frac{\sum\limits_{1}^{p}x_{j}}{p} - {\frac{\left( {l - q} \right)}{p}{k.}}}} & (6)\end{matrix}$The method convergences when|x* ^((n+1)) −x* _((n))|<δ,where δ is chosen according to the required sensitivity accuracy. Theiteration converges relatively fast for the real or actual case, whichmakes the method very practical. It can be seen from equation (6) abovethat for Gaussian distributed data x*=x⁰.

The above-described approach is relatively robust against gross errors,as well as being relatively stable against small biased data. Thisapproach may also lead to a more robust sensitivity determination, aswell as help to speed up the measurement process. Moreover, using theabove-described selection process for determining k, this may result inthe exclusion of potentially noisy points. Further, the use of a closedform equation may also contribute to fast convergence optimization, aswill be appreciated by those skilled in the art.

The bit error level data sets illustrated in FIGS. 4 and 5 demonstratethe difference between a traditional average method vs. the Huberapproach set forth above without noise and when spurious data points arepresent in the data set (i.e., with noise), respectively. Sample sets oftwenty-five measurements were used in both FIGS. 4 and 5.

For the present example, spurious data was caused by opening the door ofa shielded test box, which would not ordinarily be done during a typicaltest measurement, but is provided here to show how the two approachescan significantly differ in real world scenarios where noise is present.It can be seen that the above-described Huber approach provides a BERthat is close to the standard average BER in FIG. 4 without spuriousdata (i.e., 1.38 for Huber BER vs. 1.32 for average BER). Moreover, withnoise (FIG. 5) the Huber BER is significantly more accurate than thestandard average (i.e., 1.53 vs. 1.75).

Turning now to FIGS. 6 and 7, real world data sets taken outside (i.e.,not in an anechoic chamber) are shown (both of which also includetwenty-five sample points) for moderate and severe noise conditions,respectively. The Huber BER was 2.44 and more accurate than thecomparable average BER of 2.50 for the moderate noise environment (FIG.6). In the severe noise environment (FIG. 7), the Huber BER wassignificantly more accurate, i.e., 2.82 compared to 3.46 for the averageBER.

Another potential advantage of the Huber approach is that it can in someinstances provide more accurate results than the standard averageapproach even with less data points. One such example is illustrated inFIG. 8, in which the Huber BER was determined using twenty-five of atotal forty data points. As can be seen, the average with no noise usingall forty data points was 1.4, and the average with noise using allforty data points was 1.75. Yet, the Huber BER with noise and onlytwenty-five data points was 1.53. Accordingly, the Huber BER approachmay advantageously be used in certain embodiments with less samples,which reduces data sampling time and speeds up the measurements.

Exemplary components of a hand-held mobile wireless communicationsdevice 1000 that may be used in accordance the system 30 is furtherdescribed in the example below with reference to FIG. 10. The device1000 illustratively includes a housing 1200, a keypad 1400 and an outputdevice 1600. The output device shown is a display 1600, which ispreferably a full graphic LCD. Other types of output devices mayalternatively be utilized. A processing device 1800 is contained withinthe housing 1200 and is coupled between the keypad 1400 and the display1600. The processing device 1800 controls the operation of the display1600, as well as the overall operation of the mobile device 1000, inresponse to actuation of keys on the keypad 1400 by the user.

The housing 1200 may be elongated vertically, or may take on other sizesand shapes (including clamshell housing structures). The keypad mayinclude a mode selection key, or other hardware or software forswitching between text entry and telephony entry.

In addition to the processing device 1800, other parts of the mobiledevice 1000 are shown schematically in FIG. 10. These include acommunications subsystem 1001; a short-range communications subsystem1020; the keypad 1400 and the display 1600, along with otherinput/output devices 1060, 1080, 1100 and 1120; as well as memorydevices 1160, 1180 and various other device subsystems 1201. The mobiledevice 1000 is preferably a two-way RF communications device havingvoice and data communications capabilities. In addition, the mobiledevice 1000 preferably has the capability to communicate with othercomputer systems via the Internet.

Operating system software executed by the processing device 1800 ispreferably stored in a persistent store, such as the flash memory 1160,but may be stored in other types of memory devices, such as a read onlymemory (ROM) or similar storage element. In addition, system software,specific device applications, or parts thereof, may be temporarilyloaded into a volatile store, such as the random access memory (RAM)1180. Communications signals received by the mobile device may also bestored in the RAM 1180.

The processing device 1800, in addition to its operating systemfunctions, enables execution of software applications 1300A-1300N on thedevice 1000. A predetermined set of applications that control basicdevice operations, such as data and voice communications 1300A and1300B, may be installed on the device 1000 during manufacture. Inaddition, a personal information manager (PIM) application may beinstalled during manufacture. The PIM is preferably capable oforganizing and managing data items, such as e-mail, calendar events,voice mails, appointments, and task items. The PIM application is alsopreferably capable of sending and receiving data items via a wirelessnetwork 1401. Preferably, the PIM data items are seamlessly integrated,synchronized and updated via the wireless network 1401 with the deviceuser's corresponding data items stored or associated with a hostcomputer system.

Communication functions, including data and voice communications, areperformed through the communications subsystem 1001, and possiblythrough the short-range communications subsystem. The communicationssubsystem 1001 includes a receiver 1500, a transmitter 1520, and one ormore antennas 1540 and 1560. In addition, the communications subsystem1001 also includes a processing module, such as a digital signalprocessor (DSP) 1580, and local oscillators (LOs) 1601. The specificdesign and implementation of the communications subsystem 1001 isdependent upon the communications network in which the mobile device1000 is intended to operate. For example, a mobile device 1000 mayinclude a communications subsystem 1001 designed to operate with theMobitex™, Data TACT™ or General Packet Radio Service (GPRS) mobile datacommunications networks, and also designed to operate with any of avariety of voice communications networks, such as AMPS, TDMA, CDMA,WCDMA, PCS, GSM, EDGE, etc. Other types of data and voice networks, bothseparate and integrated, may also be utilized with the mobile device1000. The mobile device 1000 may also be compliant with othercommunications standards such as 3GSM, 3GPP, UMTS, etc.

Network access requirements vary depending upon the type ofcommunication system. For example, in the Mobitex and DataTAC networks,mobile devices are registered on the network using a unique personalidentification number or PIN associated with each device. In GPRSnetworks, however, network access is associated with a subscriber oruser of a device. A GPRS device therefore requires a subscriber identitymodule, commonly referred to as a SIM card, in order to operate on aGPRS network.

When required network registration or activation procedures have beencompleted, the mobile device 1000 may send and receive communicationssignals over the communication network 1401. Signals received from thecommunications network 1401 by the antenna 1540 are routed to thereceiver 1500, which provides for signal amplification, frequency downconversion, filtering, channel selection, etc., and may also provideanalog to digital conversion. Analog-to-digital conversion of thereceived signal allows the DSP 1580 to perform more complexcommunications functions, such as demodulation and decoding. In asimilar manner, signals to be transmitted to the network 1401 areprocessed (e.g. modulated and encoded) by the DSP 1580 and are thenprovided to the transmitter 1520 for digital to analog conversion,frequency up conversion, filtering, amplification and transmission tothe communication network 1401 (or networks) via the antenna 1560.

In addition to processing communications signals, the DSP 1580 providesfor control of the receiver 1500 and the transmitter 1520. For example,gains applied to communications signals in the receiver 1500 andtransmitter 1520 may be adaptively controlled through automatic gaincontrol algorithms implemented in the DSP 1580.

In a data communications mode, a received signal, such as a text messageor web page download, is processed by the communications subsystem 1001and is input to the processing device 1800. The received signal is thenfurther processed by the processing device 1800 for an output to thedisplay 1600, or alternatively to some other auxiliary I/O device 1060.A device user may also compose data items, such as e-mail messages,using the keypad 1400 and/or some other auxiliary I/O device 1060, suchas a touchpad, a rocker switch, a thumb-wheel, or some other type ofinput device. The composed data items may then be transmitted over thecommunications network 1401 via the communications subsystem 1001.

In a voice communications mode, overall operation of the device issubstantially similar to the data communications mode, except thatreceived signals are output to a speaker 1100, and signals fortransmission are generated by a microphone 1120. Alternative voice oraudio I/O subsystems, such as a voice message recording subsystem, mayalso be implemented on the device 1000. In addition, the display 1600may also be utilized in voice communications mode, for example todisplay the identity of a calling party, the duration of a voice call,or other voice call related information.

The short-range communications subsystem enables communication betweenthe mobile device 1000 and other proximate systems or devices, whichneed not necessarily be similar devices. For example, the short-rangecommunications subsystem may include an infrared device and associatedcircuits and components, or a Bluetooth™ communications module toprovide for communication with similarly-enabled systems and devices.

Many modifications and other embodiments will come to the mind of oneskilled in the art having the benefit of the teachings presented in theforegoing descriptions and the associated drawings. Therefore, it isunderstood that various modifications and embodiments are intended to beincluded within the scope of the appended claims.

That which is claimed is:
 1. A method for testing a radio frequency (RF)receiver comprising: measuring, in an anechoic chamber, a plurality ofbit error levels for the RF receiver at a given frequency from among aplurality of different frequencies; and applying a Huber function to themeasured bit error levels to generate a bit error ratio (BER) estimatefor the RF receiver, the Huber function being defined as:${\rho_{k}(f)} = \left\{ \begin{matrix}{f^{2}/2} & {{{if}\mspace{14mu}{f}} \leq k} \\{{k{f}} - {k^{2}/2}} & {{{if}\mspace{14mu}{f}} > k}\end{matrix} \right.$ where k is a positive constant.
 2. The method ofclaim 1 further comprising using the BER estimate to generate asensitivity for the RF receiver.
 3. The method of claim 1 wherein k iswithin a range of 0.8 to 1.4.
 4. The method of claim 1 wherein k isdefined as:${k = {2\sqrt{\frac{\sum\limits_{1}^{n}\left( {x_{i} - x_{0}} \right)}{n - 1}}}},$where x₀ is an initial bit error level and n is a total number of biterror levels.
 5. The method of claim 4 wherein the initial bit errorlevel x₀ is defined as:$x_{0} = {\frac{\overset{n}{\sum\limits_{i}}x_{i}}{n}.}$
 6. The methodof claim 1 wherein the BER estimate comprises a residual BER (RBER)estimate.
 7. The method of claim 1 wherein the RF receiver comprises aGlobal System for Mobile Communications (GSM) receiver.
 8. The method ofclaim 1 wherein the RF receiver comprises a General Packet Radio Service(GPRS) receiver.
 9. The method of claim 1 wherein the RF receivercomprises an Enhanced Data Rates for Global System for MobileCommunications (GSM) Evolution (EDGE) receiver.
 10. A method for testinga radio frequency (RF) receiver comprising: measuring, in an outdoorenvironment, a plurality of bit error levels for the RF receiver at agiven frequency from among a plurality of different frequencies; andapplying a Huber function to the measured bit error levels to generate abit error ratio (BER) estimate for the RF receiver, the Huber functionbeing defined as: ${\rho_{k}(f)} = \left\{ \begin{matrix}{f^{2}/2} & {{{if}\mspace{14mu}{f}} \leq k} \\{{k{f}} - {k^{2}/2}} & {{{if}\mspace{14mu}{f}} > k}\end{matrix} \right.$ where k is a positive constant.
 11. The method ofclaim 10 further comprising using the BER estimate to generate asensitivity for the RF receiver.
 12. The method of claim 10 wherein k iswithin a range of 0.8 to 1.4.
 13. The method of claim 10 wherein k isdefined as:${k = {2\sqrt{\frac{\sum\limits_{1}^{n}\left( {x_{i} - x_{0}} \right)}{n - 1}}}},$where x₀ is an initial bit error level and n is a total number of biterror levels.
 14. The method of claim 13 wherein the initial bit errorlevel x₀ is defined as: $x_{0} = {\frac{\sum\limits_{i}^{n}x_{i}}{n}.}$15. The method of claim 10 wherein the BER estimate comprises a residualBER (RBER) estimate.
 16. The method of claim 10 wherein the RF receivercomprises a Global System for Mobile Communications (GSM) receiver. 17.The method of claim 10 wherein the RF receiver comprises a GeneralPacket Radio Service (GPRS) receiver.
 18. The method of claim 10 whereinthe RF receiver comprises an Enhanced Data Rates for Global System forMobile Communications (GSM) Evolution (EDGE) receiver.
 19. A test systemfor testing a radio frequency (RF) receiver comprising: an anechoicchamber configured to contain the RF receiver; and test circuitrycoupled to the RF receiver and configured to measure, in the anechoicchamber, a plurality of bit error levels for the RF receiver at a givenfrequency from among a plurality of different frequencies, and apply aHuber function to the measured bit error levels to generate a bit errorratio (BER) estimate for the RF receiver, the Huber function beingdefined as: ${\rho_{k}(f)} = \left\{ \begin{matrix}{f^{2}/2} & {{{if}\mspace{14mu}{f}} \leq k} \\{{k{f}} - {k^{2}/2}} & {{{if}\mspace{14mu}{f}} > k}\end{matrix} \right.$ where k is a positive constant.
 20. The testsystem of claim 19 wherein the test circuitry is configured to use theBER estimate to generate a sensitivity for the RF receiver.
 21. The testsystem of claim 19 wherein k is within a range of 0.8 to 1.4.
 22. Thetest system of claim 19 wherein k is defined as:${k = {2\sqrt{\frac{\sum\limits_{1}^{n}\left( {x_{i} - x^{0}} \right)}{n - 1}}}},$where x₀ is an initial bit error level and n is a total number of biterror levels.
 23. The test system of claim 22 wherein the initial biterror level x₀ is defined as:$x_{0} = {\frac{\sum\limits_{i}^{n}x_{i}}{n}.}$
 24. The test system ofclaim 19 wherein the BER estimate comprises a residual BER (RBER)estimate.